# Differentially Private Gradient Flow based on the Sliced Wasserstein Distance

arXiv: https://arxiv.org/abs/2312.08227 · HTML: https://mlsanity.com/paper/2312.08227/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 62 references that are in the map.
- First posted on arXiv 13 Dec 2023.

## Builds on (most cited first, 30 of 32)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Deep Learning with Differential Privacy](https://mlsanity.com/paper/1607.00133/) (2016)
- [Membership Inference Attacks against Machine Learning Models](https://mlsanity.com/paper/1610.05820/) (2016)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Extracting Training Data from Diffusion Models](https://mlsanity.com/paper/2301.13188/) (2023)
- [Wasserstein GAN](https://mlsanity.com/paper/1701.07875/) (2017)
- [Deep Learning for Medical Image Analysis](https://mlsanity.com/paper/1708.08987/) (2017)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Differentially Private Generative Adversarial Network](https://mlsanity.com/paper/1802.06739/) (2018)
- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Differentially Private Diffusion Models](https://mlsanity.com/paper/2210.09929/) (2022)
- [GS-WGAN: A Gradient-Sanitized Approach for Learning Differentially Private Generators](https://mlsanity.com/paper/2006.08265/) (2020)
- [Differentially Private Diffusion Models Generate Useful Synthetic Images](https://mlsanity.com/paper/2302.13861/) (2023)
- [Statistical and Topological Properties of Sliced Probability Divergences](https://mlsanity.com/paper/2003.05783/) (2020)
- [DP-MERF: Differentially Private Mean Embeddings with Random Features for Practical Privacy-Preserving Data Generation](https://mlsanity.com/paper/2002.11603/) (2020)
- [Sliced-Wasserstein Flows: Nonparametric Generative Modeling via Optimal Transport and Diffusions](https://mlsanity.com/paper/1806.08141/) (2018)
- [Large-Scale Wasserstein Gradient Flows](https://mlsanity.com/paper/2106.00736/) (2021)
- [On the Reconstruction of Face Images from Deep Face Templates](https://mlsanity.com/paper/1703.00832/) (2017)
- [Don't Generate Me: Training Differentially Private Generative Models with Sinkhorn Divergence](https://mlsanity.com/paper/2111.01177/) (2021)
- [G-PATE: Scalable Differentially Private Data Generator via Private Aggregation of Teacher Discriminators](https://mlsanity.com/paper/1906.09338/) (2019)
- [From optimal transport to generative modeling: the VEGAN cookbook](https://mlsanity.com/paper/1705.07642/) (2017)
- [Pre-trained Perceptual Features Improve Differentially Private Image Generation](https://mlsanity.com/paper/2205.12900/) (2022)
- [Privacy Amplification by Mixing and Diffusion Mechanisms](https://mlsanity.com/paper/1905.12264/) (2019)
- [Efficient Gradient Flows in Sliced-Wasserstein Space](https://mlsanity.com/paper/2110.10972/) (2021)
- [Unifying GANs and Score-Based Diffusion as Generative Particle Models](https://mlsanity.com/paper/2305.16150/) (2023)
- [GAN and VAE from an Optimal Transport Point of View](https://mlsanity.com/paper/1706.01807/) (2017)
- [Differentially Private Sliced Wasserstein Distance](https://mlsanity.com/paper/2107.01848/) (2021)

## Built on by (most cited first, 4 of 4)

- [Privacy-preserving data release leveraging optimal transport and particle gradient descent](https://mlsanity.com/paper/2401.17823/) (2024)
- [Learning with Differentially Private (Sliced) Wasserstein Gradients](https://mlsanity.com/paper/2502.01701/) (2025)
- [Private Evolution Converges](https://mlsanity.com/paper/2506.08312/) (2025)
- [Liouville PDE-based sliced-Wasserstein flow](https://mlsanity.com/paper/2505.17204/) (2025)

## Abstract (arXiv)

Safeguarding privacy in sensitive training data is paramount, particularly in the context of generative modeling. This can be achieved through either differentially private stochastic gradient descent or a differentially private metric for training models or generators. In this paper, we introduce a novel differentially private generative modeling approach based on a gradient flow in the space of probability measures. To this end, we define the gradient flow of the Gaussian-smoothed Sliced Wasserstein Distance, including the associated stochastic differential equation (SDE). By discretizing and defining a numerical scheme for solving this SDE, we demonstrate the link between smoothing and differential privacy based on a Gaussian mechanism, due to a specific form of the SDE's drift term. We then analyze the differential privacy guarantee of our gradient flow, which accounts for both the smoothing and the Wiener process introduced by the SDE itself. Experiments show that our proposed model can generate higher-fidelity data at a low privacy budget compared to a generator-based model, offering a promising alternative.